| L(s) = 1 | + (−0.774 − 1.18i)2-s + (1.22 − 1.39i)3-s + (−0.799 + 1.83i)4-s + (2.99 + 0.196i)5-s + (−2.60 − 0.368i)6-s + (0.384 − 2.61i)7-s + (2.78 − 0.473i)8-s + (−0.0608 − 0.461i)9-s + (−2.08 − 3.69i)10-s + (−3.49 − 1.18i)11-s + (1.58 + 3.36i)12-s + (2.03 + 3.05i)13-s + (−3.39 + 1.57i)14-s + (3.95 − 3.95i)15-s + (−2.72 − 2.93i)16-s + (4.90 − 1.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.547 − 0.836i)2-s + (0.708 − 0.808i)3-s + (−0.399 + 0.916i)4-s + (1.34 + 0.0878i)5-s + (−1.06 − 0.150i)6-s + (0.145 − 0.989i)7-s + (0.985 − 0.167i)8-s + (−0.0202 − 0.153i)9-s + (−0.660 − 1.16i)10-s + (−1.05 − 0.357i)11-s + (0.457 + 0.972i)12-s + (0.565 + 0.845i)13-s + (−0.907 + 0.420i)14-s + (1.02 − 1.02i)15-s + (−0.680 − 0.733i)16-s + (1.18 − 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.991261 - 1.25385i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991261 - 1.25385i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.774 + 1.18i)T \) |
| 7 | \( 1 + (-0.384 + 2.61i)T \) |
| good | 3 | \( 1 + (-1.22 + 1.39i)T + (-0.391 - 2.97i)T^{2} \) |
| 5 | \( 1 + (-2.99 - 0.196i)T + (4.95 + 0.652i)T^{2} \) |
| 11 | \( 1 + (3.49 + 1.18i)T + (8.72 + 6.69i)T^{2} \) |
| 13 | \( 1 + (-2.03 - 3.05i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-4.90 + 1.31i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.478 + 0.235i)T + (11.5 + 15.0i)T^{2} \) |
| 23 | \( 1 + (-0.108 + 0.0143i)T + (22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (-0.840 + 4.22i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (5.69 + 3.28i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.144 + 2.19i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (2.23 - 5.39i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (8.48 - 1.68i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (2.90 - 10.8i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (9.14 + 3.10i)T + (42.0 + 32.2i)T^{2} \) |
| 59 | \( 1 + (-3.73 - 7.57i)T + (-35.9 + 46.8i)T^{2} \) |
| 61 | \( 1 + (-2.02 - 5.97i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-6.99 + 7.98i)T + (-8.74 - 66.4i)T^{2} \) |
| 71 | \( 1 + (9.07 - 3.75i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-8.90 + 6.83i)T + (18.8 - 70.5i)T^{2} \) |
| 79 | \( 1 + (-3.32 - 0.890i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-7.05 + 4.71i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-5.94 + 7.75i)T + (-23.0 - 85.9i)T^{2} \) |
| 97 | \( 1 - 3.82iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.66399453850664249372569824413, −9.971509295229987447188355032378, −9.189588008980473196530405757948, −8.057885113905812147854769861702, −7.55010637713620991700733391547, −6.36629361355771576742898259559, −4.92627458627629769427323205418, −3.37905864988163802703656504882, −2.26681180697979605452263601500, −1.31860586616062409484133555575,
1.86849901586184023482771019547, 3.28876537605734491553553682599, 5.21703881221783615483483508979, 5.45754456814261876317389295012, 6.63800733148431565142997590515, 8.100222988350417918480750811573, 8.633946886118606966555944709975, 9.563283619817533809587058222459, 10.05826440348280359037895317764, 10.74280497489168341876912397593
